Abstract

Integrals of monoidal Hom-Hopf algebras are introduced and the existence and uniqueness of integrals for finite-dimensional monoidal Hom-Hopf algebras are investigated first. Then integrals are applied to the Maschke type theorem for monoidal Hom-Hopf algebras controlling the semisimplicity and separability of monoidal Hom-Hopf algebras. Further, monoidal Hom-algebras are characterized with additional Frobenius property, and the question when finite-dimensional monoidal Hom-Hopf algebras are Frobenius is studied. As applications of integrals, the Maschke type theorem for Hom-smash product is given, and the Morita context in the Hom-category is constructed.

Received 19 November 2012Accepted 25 June 2013Published online 30 July 2013

Acknowledgments:

The author Liangyun Zhang would like to thank Professor Yongchang Zhu for inviting him to visit Hong Kong University of Science and Technology. Yuanyuan Chen would like to thank Professor Gabriella Böhm for her advice and help.

This work is supported by the College Special Research Doctoral Disciplines Point Fund of China (Grant No. 20100097110040), the Fundamental Research Funds for the Central Universities (Grant No. KYZ201125), and the Innovative Project of Jiangsu province for Graduate Cultivation (Grant No. CXLX12-0272).